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Introduction 1. The Buffon needle problem 2. Valuation and integral 3. A discrete lattice 4. The intrinsic volumes for parallelotopes 5. The lattice of polyconvex sets 6. Invariant measures on Grassmannians 7. The intrinsic volumes for polyconvex sets 8. A characterization theorem for volume 9. Hadwiger's characterization theorem 10. Kinematic formulas for polyconvex sets 11. Polyconvex sets in the sphere References Index of symbols Index.